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A Treatise on Trigonometric Series, Volume 1 deals comprehensively with the classical theory of Fourier series. This book presents the investigation of best approximations of functions by trigonometric polynomials.
Organized into six chapters, this volume begins with an overview of the fundamental concepts and theorems in the theory of trigonometric series, which play a significant role in mathematics and in many of its applications. This text then explores the properties of the Fourier coefficient function and estimates the rate at which its Fourier coefficients tend to zero. Other chapters consider some tests for the convergence of a Fourier series at a given point. This book discusses as well the conditions under which the series does converge uniformly. The final chapter deals with adjustment of a summable function outside a given perfect set.
This book is a valuable resource for advanced students and research workers. Mathematicians will also find this book useful.
Inhalt
Contents of Volume II
Translator's Preface
Author's Preface
Notation
Introductory Material
I. Analytical Theorems
Abel's Transformation
Second Mean Value Theorem
Convex Curves and Convex Sequences
II. Numerical Series, Summation
Series with Monotonically Decreasing Terms
Linear Methods of Summation
Method of Arithmetic Means [or (C, 1)]
Abel's Method
III. Inequalities for Numbers, Series and Integrals
Numerical Inequalities
Holder's Inequality
Minkowski's Inequality
O- and o-Relationships for Series and Integrals
IV. Theory of Sets and Theory of Functions
on the Upper Limit of a Sequence of Sets
Convergence in Measure
Passage to The Limit Under Lebesgue's Integral Sign
Lebesgue Points
Riemann-Stieltjes Integral
Helly's Two Theorems
Fubini's Theorem
V. Functional Analysis
Linear Functionals in C
1)
Convergence in Norm in the Spaces Lp
VI. Theory of Approximation of Functions by Trigonometric Polynomials
Elementary Properties of Trigonometric Polynomials
Bernstein's Inequality
Trigonometric Polynomial of Best Approximation
Modulus of Continuity, Modulus of Smoothness, and Integral Modulus of Continuity
Chapter I. Basic Concepts and Theorems in the Theory of Trigonometric Series
The Concept of a Trigonometric Series; Conjugate Series
The Complex Form of a Trigonometric Series
A Brief Historical Synopsis
Fourier Formulae
The Complex Form of a Fourier Series
Problems in the Theory of Fourier Series; Fourier-Lebesgue Series
Expansion Into a Trigonometric Series of a Function with Period 2l
Fourier Series for Even and Odd Functions
Fourier Series with Respect to the Orthogonal System
Completeness of an Orthogonal System
Completeness of the Trigonometric System in the Space L
Uniformly Convergent Fourier Series
The Minimum Property of the Partial Sums of a Fourier Series; Bessel's Inequality
Convergence of a Fourier Series in the Metric Space L2
Concept of the Closure of the System. Relationship Between Closure and Completeness
The Riesz-Fischer Theorem
The Riesz-Fischer Theorem and Parseval's Equality for a Trigonometric System
Parseval's Equality for the Product of Two Functions
The Tending to Zero of Fourier Coefficients
Fejér's Lemma
Estimate of Fourier Coefficients in Terms of the Integral Modulus of Continuity of the Function
Fourier Coefficients for Functions of Bounded Variation
Formal Operations on Fourier Series
Fourier Series for Repeatedly Differentiated Functions
on Fourier Coefficients for Analytic Functions
The Simplest Cases of Absolute and Uniform Convergence of Fourier Series
Weierstrass's Theorem on The Approximation of a Continuous Function by Trigonometric Polynomials
The Density of a Class of Trigonometric Polynomials in the Spaces Lp(P = 1)
Dirichlet's Kernel and its Conjugate Kernel
Sine or Cosine Series with Monotonically Decreasing Coefficients
Integral Expressions for the Partial Sums of a Fourier Series and its Conjugate Series
Simplification of Expressions for Sn(X) and Sn(X)
Riemann's Principle of Localization
Steinhaus's Theorem
Integral 8 0[(sinx)/x] dx. Lebesgue Constants
Estimate of the Partial Sums of a Fourier Series of a Bounded Function
Criterion of Convergence of a Fourier Series
Dini's Test
Jordan's Test
Integration of Fourier Series
Gibbs's Phenomenon
Determination of the Magnitude of the Discontinuity of a Function from its Fourier Series
Singularities of Fourier Series of Continuous Functions. Fejér Polynomials
A Continuous Function with a Fourier Series Which Converges Everywhere But Not Uniformly
Continuous Function with a Fourier Series Divergent at One Point (Fejér's Example)
Divergence at One Point (Lebesgue's Example)
Summation of a Fourier Series by Fejér's Method
Corollaries of Fejér's Theorem
Fejér-Lebesgue Theorem
Estimate of the Partial Sums of a Fourier Series
Convergence Factors
Comparison of Dirichlet and Fejér Kernels
Summation of Fourier Series by the Abel-Poisson Method
Poisson Kernel and Poisson Integral
Behaviour of the Poisson Integral at Points of Continuity of a Function
Behaviour of a Poisson Integral in the General Case
The Dirichlet Problem
Summation by Poisson's Method of a Differentiated Fourier Series
Poisson-Stieltjes Integral
Fejér and Poisson Sums for Different Classes of Functions
General Trigonometric Series. The Lusin-Denjoy Theorem
The Cantor-Lebesgue Theorem
an Example of an Everywhere Divergent Series with Coefficients Tending to Zero
A Study of the Convergence of One Class of Trigonometric Series
Lacunary Sequences and Lacunary Series
Smooth Functions
The Schwarz Second Derivative
Riemann's Method of Summation
Application of Riemann's Method of Summation to Fourier Series
Cantor's Theorem of Uniqueness
Riemann's Principle of Localization for General Trigonometric Series
Du Bois-Reymond's Theorem
Problems
Chapter II. Fourier Coefficients
Introduction
The Order of Fourier Coefficients for Functions of Bounded Variation. Criterion for the Continuity of Functions of Bounded Variation
Concerning Fourier Coefficients for Functions of the Class Lip a
The Relationship Between the Order of Summability of a Function and the Fourier Coefficients
The Generalization of Parseval's Equality for the Product of Two Functions
The Rate at Which the Fourier Coefficients of Summable Functions Tend to Zero
Auxiliary Theorems Concerning The Rademacher System
Absence of Criteria Applicable to the Moduli of Coefficients
Some Necessity Conditions for Fourier Coefficients
Salem's Necessary and Sufficient Conditions
The Trigonometric Problem of Moments
Coefficients of Trigonometric Series with Non-Negative Partial Sums
Transformation of Fourier Series
Problems
Chapter III. The Convergence of a Fourier Series at a Point
Introduction
Comparison of the Dini and Jordan Tests
The De La Vallée-Poussin Test and its Comparison with the Dini and Jordan…